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31
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What is the average of all positive integers that have four digits when written in base 3, but two digits when written in base 6? Write your answer in base 10.

 Jan 9, 2021
 #1
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+1

Smallest two digit base 6 number   106  = 610

Largest two digit base 6                  5 5 = 3510

 

Largest base 3 number w 4 digits   22223 = 80

smallest                                           1000  = 27

 

so numbers from 27 to 35    are in the set of parameters asked in the Q

    (27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 ) / 9 = .............

 Jan 9, 2021
 #2
avatar+292 
+1

base 3 numbers with 4 digits - 1000-2222 which is 27-80

base 6 numbers with 2 digits- 10-55 which is 6-35

the range is 27-35

the average is just the middle number for odd number consecutive positives

middle is 31

So the average of the positive integers, when written in base 3 that contains 4 digits but in base 6 contain 2 digits, is 31.

 Jan 9, 2021

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